In \(D-\) dimensional \(n\) th order topological states [1]}, [2]}, [3]}, [4]}, [5]}, bulk probing of the electronic band topology should play an important role, because the extended bulk-boundary correspondence is realized through gapless modes on the lower, \((D-n)-\) dimensional boundaries, characterized by codimension \(d_c=D-n\) , such as hinges (\(d_c=D-1\) ) and corners (\(d_c=D\) ) [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. These robust modes on lower-dimensional boundaries may be thought of as inherited from the parent, first-order topological state (with \(n=1\) ) upon partially gapping out its edge or surface modes (with \(d_c=1\) ), which can, in turn, yield a hierarchical ladder of HOT states [13]}. The gap is realized by a suitable domain wall mass which changes sign across
corners or hinges, thus localizing topological modes at these lower-dimensional boundaries. The reduced dimensionality of the boundary may, however, hinder the experimental detection of the gapless modes, and therefore HOT states require other means to directly probe the bulk electronic topology.
